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Hurewicz's Theorems and Renormings of Banach Spaces
Journal of Functional Analysis FU2904 journal of functional analysis 140, 142150 (1996) article no. 0102
Hurewicz's Theorems and Renormings of Banach Spaces Beno@^ t Bossard and Gilles Godefroy Equipe d 'Analyse, Universite Paris VI, Boite 186, Paris Cedex, France
and Robert Kaufman Mathematics Department, University of Illinois, Urbana, Illinois 61801 Received May 5, 1995
Let N(X) be the set of all equivalent norms on a separable Banach space X, equipped with the topology of uniform convergence on bounded subsets of X. We show that if X is infinite dimensional, the set of all locally uniformly rotund norms on X reduces every coanalytic set and, thus, is in particular non-Borel. Dually, we show the same result for the set of all continuously differentiable norms on X, under the assumption X* is separable. This provides an analogue to a classical result of Mazurkiewicz within convex analysis. 1996 Academic Press, Inc.
INTRODUCTION An equivalent norm & } & on a Banach space X is locally uniformly rotund (in short, LUR) if, whenever x # X and a sequence (x n ) in X satisfy lim 2(&x& 2 +&x n & 2 )&&x+x n & 2 =0,
n +
one has lim &x&x n &=0.
n +
An equivalent formulation, 1=&x&=&x n &=lim &(x+x n )2&, implies lim &x&x n &=0. 142 0022-123696 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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We denote N(X) the set of all equivalent norms on X. This set is topologically metrizable complete when equipped with the uniform convergence on bounded subsets of X. Recently, the topological nature of some collections of norms has been investigated: in [1], the collection of all norms on a separable space with a basis which is uniformly rotund in every direction was shown to be coanalytic non-Borel for the EffrosBorel structure, as well as the set of all weakly-LUR norms (see also [2, 3]). More recently, the third-named author showed in [8] that the set of all Ga^teaux-smooth norms reduces any coanalytic subset of a Polish space M through a continuous function in N(X). This implies of course that this set is coanalytic complete when N(X) is equipped with the EffrosBorel structure (see, e.g., [4, 13] for definition of this latter notion). The aim of this work is to show that by combining the methods of [8] with a classical topological theorem due to Hurewicz [7] and its extensions, the LUR case (Theorem 1) and the ``dual LUR'' case when X* is separable (Theorem 3) can also be obtained. A classical result of Mazurkiewicz [12] asserts that the set of all differentiable functions on [0, 1] is coanalytic non-Borel in C([0, 1]). We show here a similar result for the set of all convex continuously differentiable functions on an infinite-dimensional Banach space with separable dual (see Corollary 5). This illustrates the idea that convex functions on infinitedimensional Banach spaces and continuous functions on the real line bear a similar complexity. The notation we use is classical. We refer to [6] for renorming matters and to [9] for Hurewicz and Hurewicz-type theorems and for the descriptive set theory notions that we use. RESULTS We start with Theorem 1. Let X be a separable infinite-dimensional Banach space. Let M be a Polish space, and let A be an analytic subset of M. Then there exists a continuous map : M N(X): (i) (ii)
if t # A, (t)=& } & t is not strictly convex. if t A, (t)=& } & t is locally uniformly rotund.
Proof. Let Y be a fixed closed hyperplane of X. We write X=Y R. We may and do assume that Y is equipped with a LUR norm _} _ (see [6, Theorem II.2.6]). Since Y is infinite dimensional, the Polish space S=[ y # Y; _ y_=1]
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is not a K _ -set. Let S be a compactification of S. Since S is Polish, S is a G $ subset of S, but by the above it is not F _ in S. Hence, by Hurewicz's theorem ([7]; see [9, p. 133, Theorem 2]), there exists K/S homeomorphic to the Cantor set [0, 1] N such that (K "S) is countable and dense in K. It follows that F=K & S is a closed subset of S which is homeomorphic to N N . We have now Lemma 2. Let (S, d) be a metric space which contains a closed subset F homeomorphic to N N , and let A be an analytic subset of a Polish space (M, d $). Then there is a uniformly continuous map . from (M_S, d+d $) to [0, 1]: (i)
if t # A, then .(t, y 0 )=1 for some y 0 # F.
(ii)
if t # M"A, then .(t, y)<1 for all y # S.
Proof of Lemma 2. Let f: F A be a continuous map from F onto A. Let D(t, y) be the distance from (t, y) # M_S to the graph G= [( f ( y), y); s # F] of f, which is a closed subset of M_S. We let .(t, y)=1&(D(t, y)1+D(t, y)). If t 0 =f ( y 0 ) then .(t 0 , y 0 )=1. Conversely if .(t 1 , y 1 )=1 then D(t 1 , y 1 )=0 and, thus, (t 1 , y 1 ) # G and t 1 =f ( y 1 ). The other properties of . are clear from the definition. K We now define a subset R(t) of X as R(t)=(S_[0]) _ [ \(.(t, y) y, 1); y # S] and we let K(t)=conv(R(t)) the Minkowski functional of K(t) is denoted | } | t . Finally, we define for all ( y, s) # X &( y, s)& 2t = |( y, s)| 2t +_y_ 2.
(1)
Equation (1) clearly defines an equivalent norm & } & t =(t) on X, and the continuity of : M N(X) follows from the construction. We check now: (i) if t # A, & } & t is not strictly convex. Indeed, pick y # F such that .(t, y)=1. By definition of K(t), we have for all s # [&1, 1] |( y, s)| t =1.
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Hence for all s # [&1, 1] &( y, s)& t =- 2 and & } & t is not strictly convex. (ii) if t A, then & } & t is LUR. Let x n =( y n , s n ) and x=( y, s) in X be such that lim 2(&x n & 2t +&x& 2t )&&x n +x& 2t =0.
(2)
By (1) and a standard convexity argument (see [6, Fact II.2.3]) we have lim 2(_ y n _ 2 +_y_ 2 )&_ y n +y_ 2 =0
(3)
lim 2(|x n | 2t + |x| 2t )& |x n +x| 2t =0.
(4)
Since _ } _ is LUR, (3) implies lim _ y n &y_=0. We may and do assume that lim (s n )=s$ exists. Then lim x n =( y, s$) and we have to show that s=s$. It follows from (4) that lim |x n | t = |x| t and, thus, |( y, s)| t = |( y, s$)| t
(5)
|( y, s)| t = |( y, 0)| t .
(6)
if s= &s$; (5) implies that
The map g(s)= |( y, s)| t is convex, even, and increasing for s 0. Hence if (5) holds with s [&s$, s$], (6) follows. Thus if (5) holds with s{s$, there exists s>0 such that (6) holds. Let us check that this leads to a contradiction. Without loss of generality, we may and do assume that 0
y=lim(1&s) y n +s
\ : * .(t, z ) z .+ n j
n j
n j
j=1
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with * nj >0, j * nj =1, and _z nj _=1. Since _y_= |( y, 0)| t =1 and _} _ is LUR, (7) implies the existence of j(n): lim _ y&.(t, z nj(n) ) z nj(n)_=0.
(8)
n
Since .(t, z) # [0, 1] for all z # S, (8) implies that lim .(t, z nj(n) )=1 n
and lim _y&z nj(n)_=0, n
but since . is continuous, this implies .(t, y)=1, contradicting the fact that .(t, } ) fails to attain the value 1 since t A. K We now address the dual situation. Theorem 3. Let X be an infinite dimensional Banach space such that X* (resp. X) is separable. Let M be a Polish space, and let A be an analytic subset of M. Then there exists a continuous 4: M N(X): (i)
If t # A, then 4(t)=& } & t is not Ga^teaux-smooth.
(ii) If t A, then the dual norm 4(t)*=& } & * t is locally uniformly rotund (resp. strictly convex). Before proceeding to the proof let us observe that in case (i), 4(t)* is not strictly convex, while in case (ii), the norm 4(t) is Frechet-smooth (resp. Ga^teaux-smooth) (see [6, Chap. 2]). Proof. We fix as before a closed hyperplane Y of X. We write X=Y R and X*=Y* R. we may and do assume that Y is equipped with a LUR norm _ } _ whose dual norm _ } _* is also LUR (resp. strictly convex) (see [6, Corollary II.4.3]). We denote B Y* =[ f # Y*, _ f _* 1] and NA 1 =[ f # Y*; _ f _*=1=f ( y)=_y_ for some y # Y] and we prove for completeness the following. Claim 4 [5]. Since the norm _ } _ is LUR, the set NA 1 is G $ in (B Y* , w*). Indeed, since _} _ is in particular strictly convex, there is for any f # NA 1 a unique _( f ) # Y with __( f )_=1=( f, _( f )). The map _: (NA 1 , w*) (Y, & } &) is continuous since
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w*-lim( f n )=f O lim( f n , _( f )) =1 O lim( f n , _( f n )+_( f )) =2 O lim __( f n )+_( f )_=2 O lim __( f n )&_( f )_=0 since _} _ is LUR. If we let now O n =. [Vw*&open in B Y* ; _ } _&diam(_(V & NA 1 ))
If t # A, there is f 0 # F* such that .*(t, f 0 )=1. If t A, .*(t, f )<1 for all f # F*.
We observe now that, since .* is (d+d *)-uniformly continuous on (M_F*), it has a unique uniformly continuous extension 8* to the completion (M_K) of (M_F *). If we let 8*(t, f )=% t*( f ) # [0, 1] then the map t % * t is continuous from M to C(K) and t # A if and only if there is f 0 # F * such that % f*( f 0 )=1. We define a w*-compact subset R*(t) of X* by R*(t)=(B Y* _[0]) _ [ \(% *( t f ) f, 1): f # K] and K*(t)=conv*(R*(t)).
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Clearly, K*(t) is the unit ball of an equivalent dual norm on X* which we Finally, we define a dual norm 4*(t)=& } & t* by denote | } | *. t 2 2 2 &( f, s)& * t = |( f, s) t* +_ f _* .
(9)
Of course, 4(t) is the predual norm of 4*(t). The continuity of 4: M N(X) is straightforward. Let us check (i) and (ii): (i) Pick f # NA 1 such that % *( t f )=1. Let y # Y be such that f( y)=_y_=1. For any f # Y* and s # R, we have &( f, s)& * &( f, 0)& *=2 _ f _*. t t Hence for all y # Y, &( y, 0)& t =_ y_- 2. Since % *( t f )=1, we have for all s # [ &1, 1], |( f, s)| *=1. t Hence for all s # [ &1, 1], ( ( f, s), ( y, 0)) =1=&( f, s)& t* } &( y, 0)& t and, thus, the norm & } & t is not Ga^teaux-differentiable at ( y, 0). (ii) We first consider the case X* separable. Since _ } _* is then LUR, we can proceed along the exact same lines as is the proof of (ii) in Theorem 1, to prove that if & } & * t is not LUR, there exists f # NA 1 and s 0 >0 such that |( f 0 , 0)| *=1= |( f 0 , s 0 )| t* . t
(10)
Since t A, we have % *( t f )<1 for all f # NA 1 & K=F *. It follows that f ) f _*<1 for all f # K, since _ f _*<1 for all f # K "F *. _% *( t Let + be a probability measure on R*(t) such that ( f 0 , s 0 ) is the barycenter of +. The function F((g, s))=_ g_* is convex w*-l.s.c on X*; hence,
|
1=_ f 0 _* _ g_* d+( g). Since _% t*( f ) f _*<1 for all f # K, it follows that + is supported by B Y* _[0]; hence s 0 =0. This contradiction concludes the proof.
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A simple adaptation of this proof provides the case X separable, and _} _* strictly convex. If & } & t* is not strictly convex, there are (see [6, Proposition II.1.3]) g=( f, s){h=( f 1 , s 1 ) in X* such that 2 2 2 2(&g& * t +&h& t* )&&g+h&* t =0.
(11)
But (9), (11), and the strict convexity of _ } _* imply that f=f 1 and |( f, s)| *= |( f, s 1 )| t* {0. t Since g{h and, thus, s{s 1 , we can proceed like in the proof of (6) to find ( f 0 , s 0 ) such that (10) is satisfied, and then the end of the proof is identical. Since any uncountable Polish space M contains an analytic non-Borel subset A, Theorems 1 and 3 imply Corollary 5. Let X be a separable infinite-dimensional Banach space. Then: (1) The set of all equivalent LUR norms, the set of all equivalent strictly convex norms, the set of all equivalent Ga^teaux-smooth norms, are non-Borel subsets of N(X). (2) If, moreover, X* is separable, the set of all equivalent continuously differentiable norms is not a Bored subset of N(X). More generally, these sets cannot be obtained from the Borel subsets of N(X) through the Souslin operation. The proof is straightforward, once we observe that LUR norms are strictly convex, and norms whose dual norm is LUR are C 1 (see [6, Chap. 2]). So, Corollary 5 is the announced analogue to Mazurkiewicz's theorem. Of course, Theorems 1 and 3 imply also the expected ``complete coanalyticity'' results for the EffrosBorel structure on N(X), since it is weaker than the uniform Borel structure.
REFERENCES 1. B. Bossard, Coanalytic families of norms in Banach spaces, Illinois J. Math., to appear. 2. B. Bossard, Codage des espaces de Banach separables. Familles analytiques ou coanalytiques d'espaces de Banach, C.R. Acad. Sci. Paris 316, No. 1 (1993), 10051010. 3. B. Bossard, ``Memoire de These,'' Universite Paris VI, 1994. 4. J. P. R. Christensen, ``Topology and Borel Structure,'' North-Holland Math. Stud., Vol. 10, North-Holland, Amsterdam, 1974. 5. G. Debs, G. Godefroy, J. Saint Raymond, Topological properties of the set of normattaining linear functionals, Canad. J. Math., to appear.
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6. R. Deville, G. Godefroy, V. Zizler, ``Smoothness and Renormings in Banach Spaces,'' Pitman Monographs and Surveys, Vol. 64, Longman Ed., Harlow, 1993. 7. W. Hurewicz, Relative perfekte Teile von Punktmengen und Mengen (A), Fund. Math. 12 (1928), 78109. 8. R. Kaufman, Circulated Notes, May 1994. 9. A. Kechris and A. Louveau, Descriptive set theory and the structure of sets of uniqueness, London Math. Soc. Lecture Note Ser., Vol. 128, Cambridge Univ. Press, Cambridge, UK, 1987. 10. A. S. Kechris, A. Louveau, and W. H. Woodin, The structure of _-ideals of compact sets, Trans. Amer. Math. Soc. 301, No. 1 (1987), 263288. 11. A. Louveau and J. Saint Raymond, Borel classes and closed games: Wadge-type and Hurewicz-type results, Trans. Amer. Math. Soc. 304, No. 2 (1987), 431467. 12. S. Mazurkiewicz, Uber die Menge der differenzierbaren Funktionen, Fund. Math. 27 (1936), 244249. 13. J. Saint Raymond, La structure borelienne d'Effros est-elle standard? Fund. Math. 100 (1978), 201210.
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Hurewicz's Theorems and Renormings of Banach Spaces Beno@^ t Bossard and Gilles Godefroy Equipe d 'Analyse, Universite Paris VI, Boite 186, Paris Cedex, France
and Robert Kaufman Mathematics Department, University of Illinois, Urbana, Illinois 61801 Received May 5, 1995
Let N(X) be the set of all equivalent norms on a separable Banach space X, equipped with the topology of uniform convergence on bounded subsets of X. We show that if X is infinite dimensional, the set of all locally uniformly rotund norms on X reduces every coanalytic set and, thus, is in particular non-Borel. Dually, we show the same result for the set of all continuously differentiable norms on X, under the assumption X* is separable. This provides an analogue to a classical result of Mazurkiewicz within convex analysis. 1996 Academic Press, Inc.
INTRODUCTION An equivalent norm & } & on a Banach space X is locally uniformly rotund (in short, LUR) if, whenever x # X and a sequence (x n ) in X satisfy lim 2(&x& 2 +&x n & 2 )&&x+x n & 2 =0,
n +
one has lim &x&x n &=0.
n +
An equivalent formulation, 1=&x&=&x n &=lim &(x+x n )2&, implies lim &x&x n &=0. 142 0022-123696 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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We denote N(X) the set of all equivalent norms on X. This set is topologically metrizable complete when equipped with the uniform convergence on bounded subsets of X. Recently, the topological nature of some collections of norms has been investigated: in [1], the collection of all norms on a separable space with a basis which is uniformly rotund in every direction was shown to be coanalytic non-Borel for the EffrosBorel structure, as well as the set of all weakly-LUR norms (see also [2, 3]). More recently, the third-named author showed in [8] that the set of all Ga^teaux-smooth norms reduces any coanalytic subset of a Polish space M through a continuous function in N(X). This implies of course that this set is coanalytic complete when N(X) is equipped with the EffrosBorel structure (see, e.g., [4, 13] for definition of this latter notion). The aim of this work is to show that by combining the methods of [8] with a classical topological theorem due to Hurewicz [7] and its extensions, the LUR case (Theorem 1) and the ``dual LUR'' case when X* is separable (Theorem 3) can also be obtained. A classical result of Mazurkiewicz [12] asserts that the set of all differentiable functions on [0, 1] is coanalytic non-Borel in C([0, 1]). We show here a similar result for the set of all convex continuously differentiable functions on an infinite-dimensional Banach space with separable dual (see Corollary 5). This illustrates the idea that convex functions on infinitedimensional Banach spaces and continuous functions on the real line bear a similar complexity. The notation we use is classical. We refer to [6] for renorming matters and to [9] for Hurewicz and Hurewicz-type theorems and for the descriptive set theory notions that we use. RESULTS We start with Theorem 1. Let X be a separable infinite-dimensional Banach space. Let M be a Polish space, and let A be an analytic subset of M. Then there exists a continuous map : M N(X): (i) (ii)
if t # A, (t)=& } & t is not strictly convex. if t A, (t)=& } & t is locally uniformly rotund.
Proof. Let Y be a fixed closed hyperplane of X. We write X=Y R. We may and do assume that Y is equipped with a LUR norm _} _ (see [6, Theorem II.2.6]). Since Y is infinite dimensional, the Polish space S=[ y # Y; _ y_=1]
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is not a K _ -set. Let S be a compactification of S. Since S is Polish, S is a G $ subset of S, but by the above it is not F _ in S. Hence, by Hurewicz's theorem ([7]; see [9, p. 133, Theorem 2]), there exists K/S homeomorphic to the Cantor set [0, 1] N such that (K "S) is countable and dense in K. It follows that F=K & S is a closed subset of S which is homeomorphic to N N . We have now Lemma 2. Let (S, d) be a metric space which contains a closed subset F homeomorphic to N N , and let A be an analytic subset of a Polish space (M, d $). Then there is a uniformly continuous map . from (M_S, d+d $) to [0, 1]: (i)
if t # A, then .(t, y 0 )=1 for some y 0 # F.
(ii)
if t # M"A, then .(t, y)<1 for all y # S.
Proof of Lemma 2. Let f: F A be a continuous map from F onto A. Let D(t, y) be the distance from (t, y) # M_S to the graph G= [( f ( y), y); s # F] of f, which is a closed subset of M_S. We let .(t, y)=1&(D(t, y)1+D(t, y)). If t 0 =f ( y 0 ) then .(t 0 , y 0 )=1. Conversely if .(t 1 , y 1 )=1 then D(t 1 , y 1 )=0 and, thus, (t 1 , y 1 ) # G and t 1 =f ( y 1 ). The other properties of . are clear from the definition. K We now define a subset R(t) of X as R(t)=(S_[0]) _ [ \(.(t, y) y, 1); y # S] and we let K(t)=conv(R(t)) the Minkowski functional of K(t) is denoted | } | t . Finally, we define for all ( y, s) # X &( y, s)& 2t = |( y, s)| 2t +_y_ 2.
(1)
Equation (1) clearly defines an equivalent norm & } & t =(t) on X, and the continuity of : M N(X) follows from the construction. We check now: (i) if t # A, & } & t is not strictly convex. Indeed, pick y # F such that .(t, y)=1. By definition of K(t), we have for all s # [&1, 1] |( y, s)| t =1.
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Hence for all s # [&1, 1] &( y, s)& t =- 2 and & } & t is not strictly convex. (ii) if t A, then & } & t is LUR. Let x n =( y n , s n ) and x=( y, s) in X be such that lim 2(&x n & 2t +&x& 2t )&&x n +x& 2t =0.
(2)
By (1) and a standard convexity argument (see [6, Fact II.2.3]) we have lim 2(_ y n _ 2 +_y_ 2 )&_ y n +y_ 2 =0
(3)
lim 2(|x n | 2t + |x| 2t )& |x n +x| 2t =0.
(4)
Since _ } _ is LUR, (3) implies lim _ y n &y_=0. We may and do assume that lim (s n )=s$ exists. Then lim x n =( y, s$) and we have to show that s=s$. It follows from (4) that lim |x n | t = |x| t and, thus, |( y, s)| t = |( y, s$)| t
(5)
|( y, s)| t = |( y, 0)| t .
(6)
if s= &s$; (5) implies that
The map g(s)= |( y, s)| t is convex, even, and increasing for s 0. Hence if (5) holds with s [&s$, s$], (6) follows. Thus if (5) holds with s{s$, there exists s>0 such that (6) holds. Let us check that this leads to a contradiction. Without loss of generality, we may and do assume that 0
y=lim(1&s) y n +s
\ : * .(t, z ) z .+ n j
n j
n j
j=1
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with * nj >0, j * nj =1, and _z nj _=1. Since _y_= |( y, 0)| t =1 and _} _ is LUR, (7) implies the existence of j(n): lim _ y&.(t, z nj(n) ) z nj(n)_=0.
(8)
n
Since .(t, z) # [0, 1] for all z # S, (8) implies that lim .(t, z nj(n) )=1 n
and lim _y&z nj(n)_=0, n
but since . is continuous, this implies .(t, y)=1, contradicting the fact that .(t, } ) fails to attain the value 1 since t A. K We now address the dual situation. Theorem 3. Let X be an infinite dimensional Banach space such that X* (resp. X) is separable. Let M be a Polish space, and let A be an analytic subset of M. Then there exists a continuous 4: M N(X): (i)
If t # A, then 4(t)=& } & t is not Ga^teaux-smooth.
(ii) If t A, then the dual norm 4(t)*=& } & * t is locally uniformly rotund (resp. strictly convex). Before proceeding to the proof let us observe that in case (i), 4(t)* is not strictly convex, while in case (ii), the norm 4(t) is Frechet-smooth (resp. Ga^teaux-smooth) (see [6, Chap. 2]). Proof. We fix as before a closed hyperplane Y of X. We write X=Y R and X*=Y* R. we may and do assume that Y is equipped with a LUR norm _ } _ whose dual norm _ } _* is also LUR (resp. strictly convex) (see [6, Corollary II.4.3]). We denote B Y* =[ f # Y*, _ f _* 1] and NA 1 =[ f # Y*; _ f _*=1=f ( y)=_y_ for some y # Y] and we prove for completeness the following. Claim 4 [5]. Since the norm _ } _ is LUR, the set NA 1 is G $ in (B Y* , w*). Indeed, since _} _ is in particular strictly convex, there is for any f # NA 1 a unique _( f ) # Y with __( f )_=1=( f, _( f )). The map _: (NA 1 , w*) (Y, & } &) is continuous since
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w*-lim( f n )=f O lim( f n , _( f )) =1 O lim( f n , _( f n )+_( f )) =2 O lim __( f n )+_( f )_=2 O lim __( f n )&_( f )_=0 since _} _ is LUR. If we let now O n =. [Vw*&open in B Y* ; _ } _&diam(_(V & NA 1 ))
If t # A, there is f 0 # F* such that .*(t, f 0 )=1. If t A, .*(t, f )<1 for all f # F*.
We observe now that, since .* is (d+d *)-uniformly continuous on (M_F*), it has a unique uniformly continuous extension 8* to the completion (M_K) of (M_F *). If we let 8*(t, f )=% t*( f ) # [0, 1] then the map t % * t is continuous from M to C(K) and t # A if and only if there is f 0 # F * such that % f*( f 0 )=1. We define a w*-compact subset R*(t) of X* by R*(t)=(B Y* _[0]) _ [ \(% *( t f ) f, 1): f # K] and K*(t)=conv*(R*(t)).
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Clearly, K*(t) is the unit ball of an equivalent dual norm on X* which we Finally, we define a dual norm 4*(t)=& } & t* by denote | } | *. t 2 2 2 &( f, s)& * t = |( f, s) t* +_ f _* .
(9)
Of course, 4(t) is the predual norm of 4*(t). The continuity of 4: M N(X) is straightforward. Let us check (i) and (ii): (i) Pick f # NA 1 such that % *( t f )=1. Let y # Y be such that f( y)=_y_=1. For any f # Y* and s # R, we have &( f, s)& * &( f, 0)& *=2 _ f _*. t t Hence for all y # Y, &( y, 0)& t =_ y_- 2. Since % *( t f )=1, we have for all s # [ &1, 1], |( f, s)| *=1. t Hence for all s # [ &1, 1], ( ( f, s), ( y, 0)) =1=&( f, s)& t* } &( y, 0)& t and, thus, the norm & } & t is not Ga^teaux-differentiable at ( y, 0). (ii) We first consider the case X* separable. Since _ } _* is then LUR, we can proceed along the exact same lines as is the proof of (ii) in Theorem 1, to prove that if & } & * t is not LUR, there exists f # NA 1 and s 0 >0 such that |( f 0 , 0)| *=1= |( f 0 , s 0 )| t* . t
(10)
Since t A, we have % *( t f )<1 for all f # NA 1 & K=F *. It follows that f ) f _*<1 for all f # K, since _ f _*<1 for all f # K "F *. _% *( t Let + be a probability measure on R*(t) such that ( f 0 , s 0 ) is the barycenter of +. The function F((g, s))=_ g_* is convex w*-l.s.c on X*; hence,
|
1=_ f 0 _* _ g_* d+( g). Since _% t*( f ) f _*<1 for all f # K, it follows that + is supported by B Y* _[0]; hence s 0 =0. This contradiction concludes the proof.
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A simple adaptation of this proof provides the case X separable, and _} _* strictly convex. If & } & t* is not strictly convex, there are (see [6, Proposition II.1.3]) g=( f, s){h=( f 1 , s 1 ) in X* such that 2 2 2 2(&g& * t +&h& t* )&&g+h&* t =0.
(11)
But (9), (11), and the strict convexity of _ } _* imply that f=f 1 and |( f, s)| *= |( f, s 1 )| t* {0. t Since g{h and, thus, s{s 1 , we can proceed like in the proof of (6) to find ( f 0 , s 0 ) such that (10) is satisfied, and then the end of the proof is identical. Since any uncountable Polish space M contains an analytic non-Borel subset A, Theorems 1 and 3 imply Corollary 5. Let X be a separable infinite-dimensional Banach space. Then: (1) The set of all equivalent LUR norms, the set of all equivalent strictly convex norms, the set of all equivalent Ga^teaux-smooth norms, are non-Borel subsets of N(X). (2) If, moreover, X* is separable, the set of all equivalent continuously differentiable norms is not a Bored subset of N(X). More generally, these sets cannot be obtained from the Borel subsets of N(X) through the Souslin operation. The proof is straightforward, once we observe that LUR norms are strictly convex, and norms whose dual norm is LUR are C 1 (see [6, Chap. 2]). So, Corollary 5 is the announced analogue to Mazurkiewicz's theorem. Of course, Theorems 1 and 3 imply also the expected ``complete coanalyticity'' results for the EffrosBorel structure on N(X), since it is weaker than the uniform Borel structure.
REFERENCES 1. B. Bossard, Coanalytic families of norms in Banach spaces, Illinois J. Math., to appear. 2. B. Bossard, Codage des espaces de Banach separables. Familles analytiques ou coanalytiques d'espaces de Banach, C.R. Acad. Sci. Paris 316, No. 1 (1993), 10051010. 3. B. Bossard, ``Memoire de These,'' Universite Paris VI, 1994. 4. J. P. R. Christensen, ``Topology and Borel Structure,'' North-Holland Math. Stud., Vol. 10, North-Holland, Amsterdam, 1974. 5. G. Debs, G. Godefroy, J. Saint Raymond, Topological properties of the set of normattaining linear functionals, Canad. J. Math., to appear.
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6. R. Deville, G. Godefroy, V. Zizler, ``Smoothness and Renormings in Banach Spaces,'' Pitman Monographs and Surveys, Vol. 64, Longman Ed., Harlow, 1993. 7. W. Hurewicz, Relative perfekte Teile von Punktmengen und Mengen (A), Fund. Math. 12 (1928), 78109. 8. R. Kaufman, Circulated Notes, May 1994. 9. A. Kechris and A. Louveau, Descriptive set theory and the structure of sets of uniqueness, London Math. Soc. Lecture Note Ser., Vol. 128, Cambridge Univ. Press, Cambridge, UK, 1987. 10. A. S. Kechris, A. Louveau, and W. H. Woodin, The structure of _-ideals of compact sets, Trans. Amer. Math. Soc. 301, No. 1 (1987), 263288. 11. A. Louveau and J. Saint Raymond, Borel classes and closed games: Wadge-type and Hurewicz-type results, Trans. Amer. Math. Soc. 304, No. 2 (1987), 431467. 12. S. Mazurkiewicz, Uber die Menge der differenzierbaren Funktionen, Fund. Math. 27 (1936), 244249. 13. J. Saint Raymond, La structure borelienne d'Effros est-elle standard? Fund. Math. 100 (1978), 201210.
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